Our results lead naturally to several new questions. For example, it is well known [1]}, that if \(\Gamma \subset \mathbb {R}^n\)  is an \(m\) –dimensional Lipschitz graph and \(f \in L_1(\Gamma )\)  then the principal values of the Riesz transform \(T^{m}[f \,\mathrm {d}\nu _m] (p)\) , exist for \(\mathcal {H}^m\) –a.e. \(x \in \Gamma \) . The proof uses that \(T^m\)  is \(L_2\) –bounded
and in light of Theorem REF , it is quite unclear if the same result holds in \(\mathbb {H}\) . We do anticipate that a modification of the construction in the current paper might be used to produce an intrinsic Lipschitz graph \(\Gamma \)  such that principal values of \(T^{\mathsf {R}}[f \,\mathrm {d}\nu _{\Gamma }]\)  fail to exist \(\nu _{\Gamma }\) –a.e. for (certain) functions \(f \in L_1\) , but we will not consider this problem here.
